Hi Joe:
A quick search found an expensive part (about $6.00), an LT1152
https://www.electro-tech-online.com/custompdfs/2013/04/lt1152.pdf. Note the circuit on PDF page 1, a rail to rail buffer. Alec says the input Z is about 300K, so this will go rail to rail. With any IC, linear or digital bypass caps are installed from the power pin to ground. So you would need a 0.1 uF polyester cap from +5 to ground.
Assuming, the wiper will go from 0-5V with it disconnected, the wiper would go to the input of this OP amp wired as in the diagram. The output would go to to where the wiper was.
The OP amp which stands for (Operational Amplifier) has a very high input impeadace, so it doesn;t load the POT. It only draws about 200 pico amps from it's input. That's like 200 e-12 amps. It also has a low output impedance, so you don't get the divider effect from say 50K and 300K. It's more like <1 ohm and 300K.
If you can't get 0-5V with the wiper disconnected, then that design won't work. If you can't get 0-5V, then we can buffer the input to the POT. So, do the test first.
If you wanted to get smaller, you could use an SMT package and a small SMT breadboard. We could probably find a cheaper part. The one randomly picked out is "chopper stabilized, zero drift" which is more expensive.